From the proof of Theorem REF , we know that \( \gamma \le \rho (D) <1\)  if \( 0\le \gamma <1 \)  and \(0<\alpha < 2(1+\gamma )/\lambda _{1}\) , and \( \rho (C^{(m)}) = \rho ^M(D) + \zeta _n \) , where \( \zeta _n\rightarrow _p 0 \)  as \( n\rightarrow \infty \) .
Gelfand formula [1]} guarantees that \( \lim _{t \rightarrow \infty } \Vert  \lbrace C^{(m)} \rbrace ^t \Vert ^{1/t} = \rho (C^{(m)})\) . Then we have \( \Vert  \lbrace C^{(m)} \rbrace ^t \Vert  = (\rho (C^{(m)}) + \eta _t)^t \) , where \( \eta _t\rightarrow 0 \)  as \( t \rightarrow \infty \) . Recall that \( r^{(t,m)} = \lbrace C^{(m)}\rbrace ^t r^{(0,m)}\) . Consequently, \( \Vert \delta ^{(t, m)}\Vert  \le \Vert r^{(t,m)}\Vert  \le \Vert \lbrace C^{(m)}\rbrace ^t\Vert \ \Vert r^{(0,m)}\Vert  \le \lbrace \rho ^M(D) + \zeta _n + \eta _t\rbrace ^t \big (\Vert \delta ^{(0, m)} \Vert  + \Vert \delta ^{(0, m-1)} \Vert  \big )\) . Then the conclusions (a), (b) and (c) follow directly from Lemma REF (c), (b) and (d) in Appendix B, respectively.
